After clicking on submit Perl now starts to execute the contents of the textarea

I cannot reproduce the problem. If I run your script, HTML in the textarea is interpreted by the browser, but Perl commands are not run. To prevent the contents from evaluation, just replace all & to &amp;, and all < to &lt;.

Thanks for the links all. I'll check them out.
For now I have uploaded my sample script here: http://digioso.org/textareatest.pl
Weird thing that you cannot reproduce it. Possibly could have to do with the webserver settings or the installed Perl version then.Neither are under my control unfortunately since I'm on a large webhoster.